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## G = (C6×D4).S3order 288 = 25·32

### 11st non-split extension by C6×D4 of S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — (C6×D4).S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C12.58D6 — (C6×D4).S3
 Lower central C32 — C3×C6 — C62 — (C6×D4).S3
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for (C6×D4).S3
G = < a,b,c,d,e | a6=b4=c2=d3=1, e2=b, ab=ba, ac=ca, ad=da, eae-1=a-1b2, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=a3b2c, ede-1=d-1 >

Subgroups: 364 in 138 conjugacy classes, 57 normal (11 characteristic)
C1, C2, C2 [×3], C3 [×4], C4 [×2], C22, C22 [×4], C6 [×4], C6 [×12], C8 [×2], C2×C4, D4 [×2], C23 [×2], C32, C12 [×8], C2×C6 [×4], C2×C6 [×16], M4(2) [×2], C2×D4, C3×C6, C3×C6 [×3], C3⋊C8 [×8], C2×C12 [×4], C3×D4 [×8], C22×C6 [×8], C4.D4, C3×C12 [×2], C62, C62 [×4], C4.Dic3 [×8], C6×D4 [×4], C324C8 [×2], C6×C12, D4×C32 [×2], C2×C62 [×2], C12.D4 [×4], C12.58D6 [×2], D4×C3×C6, (C6×D4).S3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], Dic3 [×8], D6 [×4], C22⋊C4, C3⋊S3, C2×Dic3 [×4], C3⋊D4 [×8], C4.D4, C3⋊Dic3 [×2], C2×C3⋊S3, C6.D4 [×4], C2×C3⋊Dic3, C327D4 [×2], C12.D4 [×4], C625C4, (C6×D4).S3

Smallest permutation representation of (C6×D4).S3
On 72 points
Generators in S72
(1 59 17 5 63 21)(2 18 64)(3 61 19 7 57 23)(4 20 58)(6 22 60)(8 24 62)(9 25 50 13 29 54)(10 51 30)(11 27 52 15 31 56)(12 53 32)(14 55 26)(16 49 28)(33 47 70 37 43 66)(34 71 44)(35 41 72 39 45 68)(36 65 46)(38 67 48)(40 69 42)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)
(1 5)(2 6)(11 15)(12 16)(17 21)(18 22)(27 31)(28 32)(35 39)(36 40)(41 45)(42 46)(49 53)(52 56)(59 63)(60 64)(65 69)(68 72)
(1 68 56)(2 49 69)(3 70 50)(4 51 71)(5 72 52)(6 53 65)(7 66 54)(8 55 67)(9 57 33)(10 34 58)(11 59 35)(12 36 60)(13 61 37)(14 38 62)(15 63 39)(16 40 64)(17 41 27)(18 28 42)(19 43 29)(20 30 44)(21 45 31)(22 32 46)(23 47 25)(24 26 48)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)

G:=sub<Sym(72)| (1,59,17,5,63,21)(2,18,64)(3,61,19,7,57,23)(4,20,58)(6,22,60)(8,24,62)(9,25,50,13,29,54)(10,51,30)(11,27,52,15,31,56)(12,53,32)(14,55,26)(16,49,28)(33,47,70,37,43,66)(34,71,44)(35,41,72,39,45,68)(36,65,46)(38,67,48)(40,69,42), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72), (1,5)(2,6)(11,15)(12,16)(17,21)(18,22)(27,31)(28,32)(35,39)(36,40)(41,45)(42,46)(49,53)(52,56)(59,63)(60,64)(65,69)(68,72), (1,68,56)(2,49,69)(3,70,50)(4,51,71)(5,72,52)(6,53,65)(7,66,54)(8,55,67)(9,57,33)(10,34,58)(11,59,35)(12,36,60)(13,61,37)(14,38,62)(15,63,39)(16,40,64)(17,41,27)(18,28,42)(19,43,29)(20,30,44)(21,45,31)(22,32,46)(23,47,25)(24,26,48), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)>;

G:=Group( (1,59,17,5,63,21)(2,18,64)(3,61,19,7,57,23)(4,20,58)(6,22,60)(8,24,62)(9,25,50,13,29,54)(10,51,30)(11,27,52,15,31,56)(12,53,32)(14,55,26)(16,49,28)(33,47,70,37,43,66)(34,71,44)(35,41,72,39,45,68)(36,65,46)(38,67,48)(40,69,42), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72), (1,5)(2,6)(11,15)(12,16)(17,21)(18,22)(27,31)(28,32)(35,39)(36,40)(41,45)(42,46)(49,53)(52,56)(59,63)(60,64)(65,69)(68,72), (1,68,56)(2,49,69)(3,70,50)(4,51,71)(5,72,52)(6,53,65)(7,66,54)(8,55,67)(9,57,33)(10,34,58)(11,59,35)(12,36,60)(13,61,37)(14,38,62)(15,63,39)(16,40,64)(17,41,27)(18,28,42)(19,43,29)(20,30,44)(21,45,31)(22,32,46)(23,47,25)(24,26,48), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72) );

G=PermutationGroup([(1,59,17,5,63,21),(2,18,64),(3,61,19,7,57,23),(4,20,58),(6,22,60),(8,24,62),(9,25,50,13,29,54),(10,51,30),(11,27,52,15,31,56),(12,53,32),(14,55,26),(16,49,28),(33,47,70,37,43,66),(34,71,44),(35,41,72,39,45,68),(36,65,46),(38,67,48),(40,69,42)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72)], [(1,5),(2,6),(11,15),(12,16),(17,21),(18,22),(27,31),(28,32),(35,39),(36,40),(41,45),(42,46),(49,53),(52,56),(59,63),(60,64),(65,69),(68,72)], [(1,68,56),(2,49,69),(3,70,50),(4,51,71),(5,72,52),(6,53,65),(7,66,54),(8,55,67),(9,57,33),(10,34,58),(11,59,35),(12,36,60),(13,61,37),(14,38,62),(15,63,39),(16,40,64),(17,41,27),(18,28,42),(19,43,29),(20,30,44),(21,45,31),(22,32,46),(23,47,25),(24,26,48)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72)])

51 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 4A 4B 6A ··· 6L 6M ··· 6AB 8A 8B 8C 8D 12A ··· 12H order 1 2 2 2 2 3 3 3 3 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 2 4 4 2 2 2 2 2 2 2 ··· 2 4 ··· 4 36 36 36 36 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + - + image C1 C2 C2 C4 S3 D4 D6 Dic3 C3⋊D4 C4.D4 C12.D4 kernel (C6×D4).S3 C12.58D6 D4×C3×C6 C2×C62 C6×D4 C3×C12 C2×C12 C22×C6 C12 C32 C3 # reps 1 2 1 4 4 2 4 8 16 1 8

Matrix representation of (C6×D4).S3 in GL8(𝔽73)

 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 37 45 0 0 0 0 0 0 45 37 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 3 0 1 0 0 0 0 0 2 0 0 1
,
 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 71 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 3 72 3 0 0 0 0 72 2 48 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 3 0 1 0 0 0 0 0 1 0 25 72
,
 36 28 0 0 0 0 0 0 28 36 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 17 12 0 0 0 0 0 0 61 56 0 0 0 0 0 0 0 0 57 44 0 0 0 0 0 0 29 16 0 0 0 0 0 0 0 0 72 0 48 0 0 0 0 0 0 0 48 1 0 0 0 0 0 3 1 0 0 0 0 0 1 1 25 0

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,37,45,0,0,0,0,0,0,45,37,0,0,0,0,0,0,0,0,72,0,3,2,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,1,0,72,0,0,0,0,71,72,3,2,0,0,0,0,0,0,72,48,0,0,0,0,0,0,3,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,72,3,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,25,0,0,0,0,0,0,0,72],[36,28,0,0,0,0,0,0,28,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[17,61,0,0,0,0,0,0,12,56,0,0,0,0,0,0,0,0,57,29,0,0,0,0,0,0,44,16,0,0,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,3,1,0,0,0,0,48,48,1,25,0,0,0,0,0,1,0,0] >;

(C6×D4).S3 in GAP, Magma, Sage, TeX

(C_6\times D_4).S_3
% in TeX

G:=Group("(C6xD4).S3");
// GroupNames label

G:=SmallGroup(288,308);
// by ID

G=gap.SmallGroup(288,308);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,219,100,675,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^4=c^2=d^3=1,e^2=b,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1*b^2,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^3*b^2*c,e*d*e^-1=d^-1>;
// generators/relations

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